Of course, Anglicans for many years continued to read form prayers even after the Reformation, but many (if not most) of the Reformation churches quickly abandoned form prayers. Sadly, a few are bringing back form prayers, but that's a topic for another post.
This narrative builds a frequently heard straw man (link) that I think is both sadly not entirely a straw man (i.e. some people really hold to the position represented) and also not an intentional misrepresentation (i.e. the author really thinks that extemporaneous prayer advocates really justify themselves that way).
What is the straw man? The straw man is that we oppose form prayers because we want to "really mean" what we are praying. The narrative makes a compelling argument against that straw man (or is it a straw man?). We "really mean" the form psalms we sing, and we "really mean" the Scriptures that we read.
Why then the extemporaneous prayers?
1) The example of Scripture. Scripture is full of examples of prayers, and these prayers (with one notable exception) are extemporaneous, that is to say that they are ad hoc - to the occasion.
2) The notable exception is the Lord's Prayer (RCCers - we're talking about the Pater Noster or "Our Father.") But this prayer is not presented in Scripture as a form prayer to be prayed as such, but as a template for prayer. It is pray "like this" not pray "these words."
Matthew 6:9 After this manner therefore pray ye: Our Father which art in heaven, Hallowed be thy name.
Luke 11:2 And he said unto them, When ye pray, say, Our Father which art in heaven, Hallowed be thy name. Thy kingdom come. Thy will be done, as in heaven, so in earth.
(the more specific - Matthew, providing the interpretation for the more general, Luke)
The Greek word in Matthew is ουτως = thus, like this, in this way
The Greek word in Luke is λεγετε = lay out, relate
"Say" is not a very precise translation, even though it is accurate. In short, the point of both Matthew and Luke is to provide a template, with the important matters to be included in one's prayers, sometimes summarized by the mneumonic:
3) The analogy to sermons/homilies. Just as a pastor tailors the sermon or homily for the congregation, applying the truths of scripture to his flock, the man praying applies Scriptural principles of prayer (such as the template of the Lord's Prayer) to the situation at hand.
4) Last, but certainly not least, the specific Scriptural admonition against rote prayers:
Matthew 6:7 But when ye pray, use not vain repetitions, as the heathen do: for they think that they shall be heard for their much speaking.
Have you ever witnessed a Roman Catholic praying a rosary, especially a non-Latin speaking Roman Catholic praying a Latin rosary, and most especially a very experienced and devout Roman Catholic senior citizen doing so with something to be done afterwards, or while engaged in something unrelated, such as driving?
Would you like to try to tell me that those prayers are not vain repetitions just like the prayers of the heathen? They are certainly repetitious, and they certainly seem to be based on a theory that if you say the prayer a lot you will be heard.